Discussion Paper No. 539
ESTIMATING SMOOTH TRANSITION
AUTOREGRESSIVE MODELS
WITH GARCH ERRORS IN THE PRESENCE
OF EXTREME OBSERVATIONS
AND OUTLIERS
Felix Chan
and
Michael McAleer
May 2001
The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan
Estimating Smooth Transition Autoregressive
Models with GARCH Errors in the Presence
of Extreme Observations and Outliers ∗
Felix Chan and Michael McAleer
Department of Economics
University of Western Australia
April 2001
Abstract
This paper investigates several empirical issues regarding quasimaximum likelihood estimation of Smooth Transition Autoregressive
(STAR) models with GARCH errors, specifically STAR-GARCH and
STAR-STGARCH. Convergence, the choice of different algorithms for
maximising the likelihood function, and the sensitivity of the estimates
to outliers and extreme observations, are examined using daily data
for S&P 500, Heng Seng and Nikkei 225 for the period January 1986
to April 2000.
∗
The first author wishes to thank the C.A. Vargovic Memorial Award at the University
of Western Australia for financial support. The second author acknowledges the financial
support of the Australian Research Council and the Institute of Social and Economic
Research at Osaka University.
1
Introduction
Interest in non-linear time series models has increased rapidly in recent years.
In particular, regime switching models have been rather popular in the class
of non-linear models. Given the substantial research activity in analysing
time-varying volatility through GARCH process (see Engle (1982) and Bollerslev (1986)), it is also of interest to investigate regime switching models with
GARCH errors. Two of the most popular specifications in this class are the
Smooth Transition Autoregressive (STAR) - Generalised Autoregressive Conditional Heteroscedasticity (STAR-GARCH) and STAR - Smooth Transition
GARCH (STAR-STGARCH) models.
Although STAR-GARCH and STAR-STGARCH are popular and have
been used widely in forecasting (Franses, Neele and van Dijk (1998), Lundbergh and Teräsvirta (1999, 2000)), the statistical and structural properties
of these models have not yet been fully established. Furthermore, the existing diagnostic tests for these models assume consistency and asymptotic
normality, but these assumed statistical properties cannot be examined in
detail because the regularity conditions are as yet unknown. Consequently,
inferences based on these assumptions may not be valid. Moreover, information criteria such as the Akaike Information Criterion (AIC) and Schwarz
Bayesian Criterion (SBC) may not be useful for gauging the adequacy of these
models as the properties of the log-likelihood functions are also presently unknown.
The lack of knowledge of the statistical properties of these models can
cause difficulties in selecting the most efficient optimization algorithm. As
noted by Lundbergh and Teräsvirta (1999) and van Dijk, Teräsvirta and
Franses (2000a), the convergence of the Quasi-Maximum Likelihood Estimator (QMLE) is sensitive to the initial values. In fact, it is unclear as to
whether different algorithms would produce the same estimates even if the
initial values were sufficiently close to the optimised values.
This paper provides empirical evidence to show that different algorithms
produce substantially different estimates for the same model. Consequently,
the interpretation of the model can differ according to the choice of algorithm.
1
Moreover, forecast performances may also be affected. This is contrary to
the common belief that different algorithms will produce similar estimates
(Lundbergh and Teräsvirta (2000)). This paper also shows that the convergence of the QMLE for the STAR-STGARCH model depends on the choice
of transition functions.
The second part of the paper examines the effects of extreme observations
and outliers on the QMLE for the STAR-GARCH and STAR-STGARCH
models. This is of interest because STAR-type models were not designed
to accommodate extreme observations and outliers. However, these models
are often used to model financial time series, which frequently exhibit excessive kurtosis. Therefore, it is important to investigate the robustness of the
QMLE for STAR-type models in the presence of extreme observations and
outliers in order to determine how best to accommodate such data.
This paper also provides empirical evidence to show that the effects of
extreme observations and outliers on the QMLE for the STAR component in
a STAR-GARCH model depend on the choice of transition functions. The
effects of such data on the QMLE for the GARCH component are similar
to those for ARMA-GARCH, as reported in Verhoeven and McAleer (1999).
This result has not previously been investigated. Moreover, empirical evidence also suggests that the QMLE for STAR-STGARCH models is sensitive
to the presence of extreme observations and outliers.
The plan of the paper is as follows. Section 2 gives a brief outline of
recent developments for the GARCH, STAR, STAR-GARCH and STARSTGARCH models. Section 3 discusses the data. Section 4 presents a detailed discussion on various optimisation algorithms and their effects on the
QMLE for STAR-GARCH models. Section 5 investigates the effects of extreme observations and outliers on the estimates for both STAR-GARCH
and STAR-STGARCH models. Section 6 gives some concluding remarks.
2
Models
This section provides a brief discussion of recent developments in modeling
GARCH, STAR, STAR-GARCH and STAR-STGARCH. Model definitions,
2
characterisitics and statistical properties will be discussed, with an emphasis
on the importance of deriving the structural and statistical properties of the
models.
2.1
ARCH/GARCH
Consider the ARMA(r, s) model:
yt =
r
X
αi yt−i +
i=1
s
X
βi εt−i + εt
(2.1)
i=1
in which εt is said to follow an Autoregressive Conditional Heteroscedasticity,
ARCH(p), process if
p
εt = ηt ht
(2.2)
where
ηt ∼ i.i.d.(0, 1)
p
X
ht = ω0 +
αi ε2t−i .
(2.3)
(2.4)
i=1
This model was proposed by Engle (1982) to relax the traditional assumption
of a constant one-period forecast variance. Engle showed that this model has
a constant unconditional variance but a non-constant variance conditional
on the past.
Engle (1982) showed that εt is second-order stationary (that is, the second moment of εt is finite) if and only if all the roots of the characteristic
polynomial
p
X
αi z i ) = 0
(1 −
i=1
lie outside the unit circle. It was assumed that the process εt starts infinitely
far in the past, with finite 2mth moment. This assumption is clearly not
possible to check in practice. However, Engle (1982) also derived the regularity condition for the existence of the moments for ARCH(1), specifically
the 2mth moment exists if and only if
αm
1
m
Y
j=1
(2j − 1) < 1.
3
Milhøj (1985) avoided Engle’s assumption and showed that εt is secondorder stationary if and only if
p
X
αi < 1.
(2.5)
i=1
He also derived the regularity condition for the existence of moments without
the restrictive assumption. Milhøj’s result is identical to Engle’s in the case
of ARCH(1) with normal ηt , but cannot be given an explicit form in the case
of ARCH(p) and m > 2.
It is interesting to note that equation (2.5) is not a necessary condition for
the strict stationarity of the ARCH(p) model. The necessary and sufficient
condition for the strict stationarity of ARCH(p) was derived by Bougerol and
Picard (1992).
Engle (1982) suggested two possible methods for estimating the parameters in equations (2.1) and (2.4) namely, the Least Squares Estimator (LSE)
and the Maximum Likelihood Estimator (MLE). The LSE is given as
Pn
ε̃t−1 ε̃t
(2.6)
δ̂ = Pnt=2
0
t=2 ε̃t−1 ε̃t−1
where δ̂ = (ω0 , α1 , ...., αp ) and ε̃t = (1, ε2t , ..., ε2t−p+1 ). Weiss (1986) and Pantula (1989) showed that δ̂ is consistent and asymptotic normal if
E(ε8t ) < ∞
which is a rather strong condition.
The conditional log-likelihood function of (2.1) given observations εt , t =
1, .., T , can be written as:
T
1 ε2t
1X 1
l(δ) =
− ln ht −
T t=1 2
2 ht
(2.7)
so that the MLE is given as
δ̂ = argmaxδ∈Θ l(δ)
assuming that δ ∈ Θ, a compact subset of Rp+1 . Engle (1982) showed that
the information matrix of this function is block-diagonal, which implies the
4
parameters in the conditional mean and the conditional variance can be estimated separately without loss of asymptotic efficiency. The estimated errors
given by the estimated conditional mean equation can be used to estimate
the equation of the conditional variance. However, it is important to note
that the restrictions ω0 > 0, αi > 0 for all i = 1, .., p, are required to ensure
that the conditional variance is positive.
Moreover, the MLE is referred to as the Quasi MLE (QMLE) when ηt
is not normal. Weiss (1986) and Pantula (1989) showed that the QMLE is
consistent and asymptotic normal if
E(ε4t ) < ∞.
This result was extended by Ling and McAleer (1999b), who showed that
the QMLE is consistent and asymptotic normal if
E(ε2t ) < ∞.
The Berndt, Hall, Hall, Hausman (1974) algorithm (BHHH) is often used to
determine δ̂ but, as suggested by Mak, Wong and Li (1997), this algorithm
has convergence problems if the initial values are not sufficiently close to the
final solutions. In such cases, a Newton-Raphson procedure should be used.
Bollerslev (1986) extended ARCH by including the lags of the conditional
variance to yield the GARCH(p, q) model, namely
ht = ω +
p
X
αi ε2t−i
i=1
+
q
X
βi ht−i .
(2.8)
i=1
If βi = 0, for all i, then GARCH(p, q) reduces to an ARCH(p) model. All
the mathematical and statistical properties of GARCH hold for ARCH, in
general, except for one case which will be discussed below.
The necessary and sufficient condition for the second-order stationarity
of (2.8) was established by Bollerslev (1986) as
p
X
i=1
αi +
q
X
i=1
5
βi < 1.
Nelson (1990) derived the necessary and sufficient condition for strict stationarity and ergodicity for GARCH(1,1) as
E(ln(α1 ηt2 + β1 )) < 0.
This condition allows α1 + β1 to be slightly larger than 1, in which case
the variance is not finite (i.e. E(ε2t ) = ∞). Note that this condition holds
for GARCH(1,1) but not for ARCH(1), that is, E(ln(α1 ηt2 )) < 0 does not
ensure strict stationarity and ergodicity for ARCH(1), because the condition
is derived under the assumption that β1 6= 0. Moreover, this condition is not
easy to apply in practice as it is the mean of an unknown random variable,
and also involves unknown parameters.
For GARCH(p, q), the necessary and sufficient condition for strict stationarity and ergodicity was established by Bougeral and Picard (1992) and
Nelson (1990). The necessary and sufficient condition for the existence of the
2mth moment of the GARCH(1,1) model was provided by Bollerslev (1986),
who also provided the necessary and sufficient condition for the fourth-order
moments of the GARCH(1,2) and GARCH(2,1) models. He and Teräsvirta
(1999a) obtained the moment conditions of a family of GARCH(1,1) models
using a similar method as in Bollerslev (1986). Ling and McAleer (1999c)
derived the sufficient condition for the existence of the stationary solution of
this family of GARCH(1,1) models, showed that He and Terasvirta’s (1999a)
condition was necessary but not sufficient, and provided the sufficient condition. He and Teräsvirta (1999b) examined the fourth moment structure of
the general GARCH(p, q) process. In the case of GARCH(1,1), the fourth
moment condition is given by
(α1 + β1 )2 + 2α12 < 1.
(2.9)
Ling (1999) obtained a sufficient condition for the existence of the 2mth
moment for the GARCH(p, q) model, based on Theorem 2.1 in Ling and Li
(1997) and Theorem 2 in Tweedie (1988). The sufficient condition is given
as
ρ[E(A⊗m
t )] < 1
6
where ρ(A) = max{eigenvalues of a matrix A}, and At is given by:


...
αp ηt
β1 ηt
...
βq ηt
α1 ηt




I(p−1)×(p−1) O(p−1)×1
O(p−1)×q


At = 
...
αp
β1
...
βq 

 α1
O(q−1)×p
I(q−1)×(q−1) O(q−1)×1
Unlike Bollerslev (1986) and He and Teräsvirta (1999a,b), Ling’s method
does not assume that the GARCH(p, q) process starts infinitely far in the
past with finite 2mth moment, and has a far simpler form than Milhøj’s
(1985) result. This condition is also necessary for the existence of the 2mth
moment, as demonstrated by Ling and McAleer (1999a). Thus, the moment
structure of a general GARCH(p, q) has been completely established. As an
extension of the GARCH(p, q) process, Ling and McAleer (1999a) derived
the necessary and sufficient moment conditions of the asymmetric power
GARCH(p, q) model of Ding et al. (1993).
The parameters in a GARCH(p, q) model are often estimated by MLE,
or by QMLE when the normality of ηt is not assumed. The log-likelihood
function of GARCH(p,q) is identical to that of (2.7), except for the definition
of ht . Naturally, ht in this case follows the definition of (2.8) instead of (2.4).
For GARCH(1,1), Lee and Hansen (1994) and Lumsdaine (1996) showed that
the QMLE is consistent and asymptotic normal if
E[ln(α1 ηt2 + β1 )] < ∞,
β 6= 0.
Ling and Li (1997) showed that the local QMLE for GARCH(p, q) is consistent and asymptotic normal if
E(ε4t ) < ∞.
For the global QMLE, Ling and McAleer (1999a) showed that
E(ε2t ) < ∞
is sufficient for consistency, and
E(ε6t ) < ∞
7
is sufficient for asymptotic normality.
The QMLE is often more efficient than LSE for ARMA-GARCH(p, q)
models. This result was first observed by Engle (1982) through a simple
fixed design regression model with an ARCH(1) process. Pantula (1989)
also showed that the MLE is more efficient than LSE for an AR model with
ARCH(1) errors. The QMLE is efficient only if ηt is normal. When ηt is not
normal, adaptive estimation is useful to obtain efficient estimators. Some
useful references include Bickel (1982), Robinson (1988) and Stoker (1991).
This estimation method is not yet available in most econometric software
packages due to its computational complexity, which explains in part the
popularity of MLE in this literature.
It is important to note that the choice of lag length in the conditional
variance equation has not been well investigated in the literature. Engle
(1982) proposed an LM test for ARCH effects, and used the test to decide
the appropriate lag length. Bollerslev (1986) used a similar test to decide the
lag length of GARCH in an empirical example, but admitted that his choice
was arbitrary. Some researchers choose the lag length for their models based
on model adequacy, using criteria such as the AIC and SBC, while others
choose their models based on in-sample forecast performance.
A distinct characteristic of GARCH-type models is their ability to capture volatility clustering. If the shock from the previous period is high (low),
the large (small) value of ε2t−1 will then influence ht . The GARCH model
can also be fitted to leptokurtic financial data and can be adapted for conditional Student t-distributed (GARCH-t) errors. The GARCH model also
offers computational advantages over extended versions thereof. For example, the log-likelihood function of GARCH is relatively simple. However,
there are several deficiencies in the linear GARCH model, as observed by
Nelson (1991). First, it is an empirical regularity that the impact of a large
negative shock is greater than a large positive shock, but a small positive
shock has a larger impact than a small negative shock. This type of asymmetric behaviour cannot be captured by the symmetric GARCH model as
the conditional variance is a function only of past squared errors and past
conditional variances.
8
Moreover, it is important to impose a restriction on all of the parameters in GARCH models to ensure the positivity of the conditional variances.
These restrictions can create difficulties in estimating GARCH, especially
when the data exhibit extreme observations and outliers.
2.2
STAR, STAR-GARCH
and STAR-STGARCH
Non-linear time series models have become very popular in recent years.
Regime switching models are very popular in the class of non-linear models,
so it is of interest to investigate regime switching models with GARCH errors.
Regime switching models will be discussed here, with emphasis on Smooth
Transition Autoregressive (STAR) models.
Tong (1978) and Tong and Lim (1980) proposed the Threshold Autoregressive (TAR) model. The TAR model assumes that the regimes switch
from one to another, as determined by the threshold variables, st , relative to
the threshold value, c. Consider the two regime case:
yt = (φ10 +
r
X
i=1
where
φ1i yt−i )(1−I(st −c))+(φ20 +
I(st − c) =
r
X
i=1
φ2i yt−i ))I(st −c)+εt (2.10)
½
0, st < c
1, st ≥ c.
Model (2.10) can also be written as:
P
½
φ10 + ri=1 φ1i yt−i st < c
yt =
P
φ20 + ri=1 φ2i yt−i st ≥ c.
The threshold variable, st , is usually (but not always) defined as a linear
combination of the lagged values of yt , that is,
st =
k
X
πi yt−i
i=0
which is often referred to as a Self Exciting TAR (SETAR). van Dijk, Teräsvirta
and Frances (2000a) relaxed this definition of threshold variables to include
non-linear combinations of the lags of yt and other exogenous variables.
9
Equation (2.10) is similar to a standard model of structural change, apart
from the definition of threshold variable and threshold value, which assumes
that the regimes switch from one to another instantly. To allow for a smooth
transition, Teras̈virta (1994) proposed the Smooth Transition Autoregressive
(STAR) model:
yt = (φ10 +
r
X
i=1
φ1i yt−i )(1 − G(st ; γ, c)) + (φ20 +
r
X
φ2i yt−i ))G(st ; γ, c) + εt
i=1
(2.11)
in which G(st ; γ, c) is the transition function, assumed to be twice differentiable, ranging from 0 to 1, and γ is the rate of transition.
Although two regimes will suffice in many empirical cases, it is straightforward to extend (2.10) to more than two regimes. Denoting φi = (φi1 , ...φir )
and xt = (yt−1 , ..., yt−r )0 , equation (2.10) can be rewritten as
yt = (φ1 xt )(1 − I(st − c) + (φ2 xt )I(st − c) + εt .
An m-regime TAR (or Multiple Regime TAR, MRTAR) model can be written
as
m
X
yt =
φi xt (I(st − ci−1 ) − I(st − ci )) + εt
(2.12)
i=1
where c1 < c2 < .... < cm , and
I(st − ci ) =
½
0, st < i or i = m
1, st ≥ i or i = 0.
For any st ∈ [ci−1 , ci ), yt = φi xt for all i = 1...m. Therefore, the regime is
determined by the threshold variable, st , relative to the threshold value, ci .
To incorporate the idea of smooth transition in equation (2.10), replace the
function I(st − ci ) in (2.12) with the transition function G(st ; γ, ci ) for all i,
yielding
yt =
m
X
i=1
φi xt (Gi−1 (st ; γi−1 , ci−1 ) − Gi (st ; γi , ci )) + εt
(2.13)
where Gi (st ; γi , ci ) is assumed to be a twice differentiable function ranging
from 0 to 1, G0 = 1 and Gm = 0. Equation (2.13) is known as the Multiple
Regime Smooth Transition Autoregressive (MRSTAR) model.
10
An extension of the basic model permits the parameter vector φi to change
over time, which is known as the Time Varying STAR (TV-STAR) model
(van Dijk, Teräsvirta and Franses (2000b)).
There are many choices of transition function, with the most popular
being the first-order logistic function:
G(st ; γ, c) =
1
1 + exp(−γ(st − c))
with the following properties:
lim G(st ; γ, c) → 0
st →−∞
lim G(st ; γ, c) → 1
st →∞
1
2
lim G(st ; γ, c) → 0
G(st ; 0, c) =
γ→−∞
lim G(st ; γ, c) → 1.
γ→∞
A STAR model with a logistic transition function is the Logistic STAR
(LSTAR). Although the logistic function is used frequently, other choices
include the Exponential STAR (ESTAR) model:
G(st ; γ, c) = 1 − exp(−γ(st − c)2 ),
γ>0
and the nth -order LSTAR:
n
Y
G(st ; γ, c) = (1 + exp(−γ (st − ci )))−1 ,
i=1
γ > 0,
ci < cj
∀i < j.
In order to use this model effectively, it is important to choose the appropriate
transition function and threshold variable. There exist many LM-type tests
to determine the appropriate choice of G(st ; γ, c) and st (a comprehensive
survey of the modelling strategy under the STAR framework is given in van
Dijk, Teräsvirta and Franses (2000a)).
Generally, the modelling cycle starts with a test of parameter constancy,
such as testing whether STAR is more appropriate than a simple linear AR
11
model. Assuming that LSTAR with two regimes is the preferred model, a
test of parameter constancy is given by
HA0 : φ11 = φ21 ,
φ12 = φ22 .
Parameters within the transition function, γ and c, are not involved in the
null hypothesis, yielding unidentified nuisance parameters. Consider the null
hypothesis of linearity as a test of
HB0 : γ = 0
in which
G(st ; 0, c) =
1
2
so that the STAR model can be written as
yt =
φ11 + φ21 φ12 + φ22
+
yt−1 + εt
2
2
which is linear, regardless of the truth of HA0 . Thus, it is important to include
parameters in the transition function for purposes of testing. This problem
can be avoided by expressing the transition function by its Taylor expansion
around γ = 0, which is a simple but important technique for hypothesis
testing with STAR-type models.
When the transition function and the threshold variable have been determined, the parameters in STAR can be estimated by Non-linear Least
Square (NLS). If
yt = F (xt ; φ) + εt
the NLS estimator is given by
T
T
X
X
2
φ̂ = argminφ
(yt − F (xt ; φ)) = argminφ
ε2t .
t=1
t=1
If εt is normal, NLS is equivalent to MLE, otherwise NLS can be interpreted
as QMLE. Wooldridge (1994) and Pötscher and Prucha (1997) demonstrated
that the NLS is consistent and asymptotic normal under certain regularity
conditions.
12
STAR models, especially LSTAR models, have been successfully applied
in a number of areas. Teräsvirta and Anderson (1992) and Teräsvirta,
Tjøstheim and Granger (1994) characterised the different dynamics of industrial production indexes for various OECD countries during expansions
and recessions using LSTAR models. Moreover, Lundbergh and Teräsvirta
(2000) examined the forecast performances of the LSTAR model for unemployment rates in Denmark and Australia, arguing that many unemployment
rates exhibit asymmetries in that the rate of increase is often higher than
the rate of decrease. Their results showed that the STAR model is superior
to its AR counterpart.
A STAR-GARCH model allows εt in equation (2.13) to follow a GARCH
process, as defined in (2.8). This natural extension has not yet been investigated widely. Lundbergh and Teräsvirta (1999) give a comprehensive
exposition of this model, but do not provide any statistical properties or regularity conditions for the existence of its moments and for stationarity. These
important properties have not yet been established. However, as the information matrix of the log-likelihood function of STAR-GARCH is block diagonal,
the parameters in the conditional mean and conditional variance equations
can be estimated separately, as in the case of ARMA-GARCH. Therefore,
the general GARCH properties described earlier are also expected to hold
for this model.
A further extension of the STAR-GARCH model is to incorporate the
concept of regime switching in the GARCH component, resulting in the
STAR-Smooth Transition GARCH (STAR-STGARCH) model. Let θi =
(θi0 , ..., θi(p+q) ), Γt = (1, ε2t−1 , ε2t−2 , ...ε2t−p , ht−1 , ..., ht−q )0 and Hi be a twice
differentiable function for all i > 0, with H0 = 1 and Hm = 0. Denote a new
threshold variable as
k
X
rt =
ζi εt
i=1
with threshold values di ∈ R for all i. Then the STAR-STGARCH model is
the same as equation (2.13), with
εt = ηt
13
p
ht
where
m
X
ht =
(θi Γt )(Hi−1 (rt ; ξi−1 , di−1 ) − Hi (rt ; ξi , di ).
(2.14)
i=1
The choice of Hi is similar to that of Gi , but is not restricted to be like Gi
in the most general case.
STAR-STGARCH is novel and has a number of distinct characteristics.
First, it is non-linear, not only in the conditional mean, but also in the conditional variance. The GARCH component is useful for capturing volatility
clustering, while the threshold variables and threshold values are useful if
the data exhibit regime switching behaviour for varying yt and εt . STARSTGARCH also exhibits asymmetries as it can be represented by setting the
threshold value to 0. Consider a simple two-regime case, with rt = εt , d = 0
and
1
H(εt ; ξ, 0) =
1 + exp(−ξ(εt − 0))
so that equation (2.14) can be rewritten as
ht = (θ1 Γt )(1 − H(εt ; ξ, 0)) + (θ2 Γt )H(εt ; ξ, 0)).
Thus, in the extreme cases where εt → −∞ and εt → ∞,
ht = (θ1 Γt )
ht = (θ2 Γt )
respectively. Therefore, the first regime is associated with εt < 0 and the
second regime with εt ≥ 0.
Although this model is potentially useful for data that exhibit non-linearity
and threshold behaviour, there are as yet no results as to the moment structure or statistical properties of the MLE. Furthermore, as asymmetric behaviour is permitted, the information matrix for this model is no longer block
diagonal and the two stage estimation method is no longer valid. It is also important to note that, as observed in van Dijk, Teräsvirta and Franses (2000a),
the MLE for both STAR-GARCH and STAR-STGARCH is extremely sensitive to initial values. The choice of algorithm in approximating the optimal
14
solution is also crucial in terms of convergence. These problems may be resolved by understanding the statistical properties and moment structure of
the model.
There are several LM-type specification tests to analyse the most appropriate model. Lundbergh and Teräsvirta (1999) and van Dijk, Teräsvirta
and Frances (2000a) provide a list of such tests. However, these tests are
based on the assumption that the model is stationary and ergodic, an assumption which cannot be checked, in general, as no regularity conditions
are as yet available. Therefore, any empirical examples using STAR-GARCH
and STAR-STGARCH remain questionable in terms of their reliability and
stability.
There is as yet no theoretical result regarding the stationarity of the
STAR, STAR-GARCH or STAR-STGARCH models. Consider a two regime
STAR model, as defined in (2.13), which can be rewritten as
yt =(φ11 (1 − G(st ; γ, c)) + φ21 G(st ; γ, c)) + (φ12 (1 − G(st ; γ, c))
+ φ22 G(st ; γ, c))yt−1 + εt .
Does the condition
φ12 (1 − G(st ; γ, c)) + φ22 G(st ; γ, c) < 1
ensure stationarity? This is not at all obvious as the transition function
G(st ; γ, c) is a function of the endogenous variable yt . More research in establishing the statistical properties and regularity conditions of these models
is required before these models can be used with confidence.
Evaluating forecast performance is also problematic. As noted by van
Dijk, Teräsvirta and Franses (2000a), even though non-linear time series often
capture certain characteristics of the data better than do linear models, the
forecast performance of the former is not always superior, and is sometimes
even worse. Clements and Hendry (1998) and Diebold and Nason (1990)
discuss various reasons for this phenomenon.
15
3
Data
Following Lundbergh and Teräsvirta (1999), all regimes for each model estimated below are assumed to follow an AR(1) process, εt is assumed to be
GARCH(1,1), st is set equal to yt−1 , and rt is set equal to εt−1 .
All models are estimated using three different stock indices, namely Standard and Poor’s 500 Composite Index (S&P), Heng Seng Index and Nikkei
225 Index. The data were obtained through the DataStream database service
and the sample is from 1/1/1986 to 11/4/2000, giving a total of 3725 data
points for each index.
Of primary concern are stock returns, Rt , which are calculated as
Rt =
Yt − Yt−1
Yt−1
where Yt denotes the index at time t.
The returns for S&P 500, Heng Seng and Nikkei 225 are given in the
following figures:
Figure 1: S&P Returns
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
0
500
1000
1500
2000
16
2500
3000
3500
4000
Figure 2: Heng Seng Returns
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0
500
1000
1500
2000
2500
3000
3500
4000
3000
3500
4000
Figure 3: Nikkei 225 Returns
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
0
500
1000
1500
2000
17
2500
S&P appears to be less volatile than either Nikkei 225 or Heng Seng, especially during the early to late 1990’s (observations 1500 to 3000) before the
Asian economic and financial crises. S&P also has fewer extreme observations
and outliers than Nikkei and Heng Seng.
Nikkei seems to have more positive shocks than S&P and Heng Seng. Although Nikkei seems more volatile, the volatility is relatively low compared
with Heng Seng, which seems to be the most volatile. There are some obvious outliers and extreme observations for all three indices. Heng Seng also
appears to have the highest number of outliers.
An obvious similarity among the three indices is the enormous decrease
in returns at observation 474, which corresponds with the share market crash
in October 1987. This is also the most significant outlier in the three indices.
The second largest decrease in returns is observation 894 for Heng Seng,
which corresponds with the Tianenman Square incident in Beijing on 4 June
1989.
4
Optimisation Algorithms
Estimation for STAR-type models is problematic as their novelty means that
existing econometric software packages do not yet have appropriate algorithms programmed. STAR can be estimated by Non-linear Least Squares
(NLS) (see van Dijk, Teräsvirta and Franses (2000a)). STAR-GARCH can
be estimated by a two-stage procedure, which involves estimating STAR
by NLS, then using the residuals to estimate GARCH by QMLE. However,
this procedure is not appropriate for STAR-STGARCH, for which the information matrix is not block diagonal, so the estimates have to be obtained
simultaneously. It is worth noting that NLS is equivalent to MLE under
the assumption of normality. If this assumption does not hold, then NLS is
equivalent to QMLE.
In practice, estimation of STAR and STAR-GARCH models can be problematic. An attempt was made in EViews to estimate LSTAR by NLS with
st = Yt−1 for S&P 500, Heng Seng and Nikkei, but was unsuccessful because
of computational problems in calculating the near singular Hessian matrix.
18
A GAUSS version 3 program was used to estimate various STAR-type
models by optimising the respective likelihood functions. Several attempts
were made to estimate LSTAR and ESTAR for S&P, Heng Seng and Nikkei,
for which the estimates of the variance did not converge. This suggests three
possibilities: (i) the variance is not constant, so that STAR-GARCH should
be used; (ii) the use of alternative optimisation algorithms (see for example,
Luenberger (1989)); and (iii) the use of alternative initial values.
In order to investigate the effects of different algorithms on the estimates
of STAR-GARCH models, two were chosen to maximise the log-likelihood
function of a LSTAR-GARCH model, as defined in equations (2.13) and (2.8),
using S&P, Heng Seng and Nikkei data. The two algorithms used are the
Newton method and Broyden-Fletcher-Goldfarb-Shanno method (BFGS), a
quasi-Newton method, with the same initial values, yielding the estimates in
Table 1 (one asterisk denotes significance at the 5% level and two asterisks
denote significance at 1%).
19
Table 1: LSTAR-GARCH Estimates
S&P
Newton
φˆ11
φˆ12
φˆ21
φˆ22
γ̂
ĉ
ω̂
α̂
β̂
Heng Seng
BFGS
1.7833** -3.0150**
1.1204** -4.1005**
-0.2862** 3.0088**
-1.3476** 3.8090**
2.4515** 1.4818**
0.8634** -0.0402**
1.17e-6
1.17e-6**
0.0753** 0.0754**
0.9166** 0.9166**
Newton
-0.2238**
2.0523**
0.2260**
-1.8170**
4.3080**
-0.0162**
7.08e-6**
0.1488**
0.8394**
Nikkei
BFGS
Newton
BFGS
0.0142
-0.2796** -1.5166**
-4.9601** -0.1930** 0.1396**
0.0007
0.0013** 1.5131**
0.3180**
0.0126
-0.2892**
3.9604** 1.6278** 1.0020**
-0.8311** -2.0104** -0.0586**
7.10e-6** 3.2e-6** 3.21e-6**
0.1486** 0.1475** 0.1465**
0.8394** 0.8519** 0.8529**
As shown in Table 1, the two algorithms produce different estimates for
the conditional mean but not of the conditional variance. In fact, a similar
set of estimates for the conditional variance can be obtained by estimating
a simple AR(1)-GARCH(1,1) model using the Berndt-Hall-Hall-Hausman
(BHHH) algorithm. This suggests that only the estimates of the conditional
mean are sensitive to the choice of algorithm, which is contrary to the findings
in Lundbergh and Teräsvirta (2000).
In particular, the high threshold values for Heng Seng imply that the
first regime dominates the second. However, since BFGS gives substantially
different estimates from Newton, it is difficult to determine which set of
estimates will produce better forecasts. Moreover, the estimates from the
two algorithms are highly significant, which makes interpretation problematic. As robustness to the choice of algorithm is not in evidence here, this
stresses the importance of establishing regularity conditions for consistency
and asymptotic normality.
A possible explanation of these differences is that the Hessian matrix of
the log-likelihood function of the LSTAR-GARCH model is not accurately
approximated by BFGS, in which case the covariance matrix will also be
unreliable. Another possibility is that there exists more than one optimum
20
for the log-likelihood function, or that it is flat. This is supported by the
similar mean likelihood scores of the two algorithms, as shown in Table 2.
Table 2: Maximum Likelihood Score
Index
Newton
BFGS
S&P
Heng Seng
Nikkei
4.24018
3.75686
3.92471
4.24018
3.75664
3.92487
As the structural and statistical properties of the STAR-GARCH models have
not yet been established, it is difficult to provide a clear and unambiguous
explanation for this result.
5
Extreme Observations and Outliers
The technical definitions of extreme observations and outliers are somewhat
arbitrary. Extreme observations are often referred to as being 2 to 3 standard deviations from the mean. Outliers are often defined as being more
than 3 standard deviations from the mean. The difference between extreme
observations and outliers is that outliers can also be defined as observations
that were not generated from the same population as the other observations
in the sample.
Stock returns often contain more extreme observations and outliers as
compared with a normal distribution. Consequently, the distribution seems
to have fatter tails, or excessive kurtosis, than a normal distribution.
In order to examine the effects of extreme observations and outliers on the
estimates for LSTAR-GARCH, ESTAR-GARCH and ESTAR-LSTGARCH,
each of the data sets is adjusted using the following trimming algorithm:
1. Calculate the standard deviation for the sample;
2. If an observation is 4 times larger than the standard deviation, it is
reduced to 4 times the standard deviation;
21
3. If an observation is between 3 and 4 times larger than the standard
deviation, it is reduced to 3 times the standard deviation;
4. If an observation is between 2.5 and 3 times larger than the standard
deviation, it is reduced to 2.5 times the standard deviation;
5. Repeat steps 1 to 4 above for every observation in the sample.
An LSTAR-GARCH model, as defined in equations (2.8) and (2.13), is estimated using both the adjusted and unadjusted S&P, Heng Seng and Nikkei
data. The Newton algorithm is used in each case. The estimates can be
found in the following Table 3.
Table 3: LSTAR-GARCH Estimates for Adjusted and Unadjusted Data
S&P
φˆ11
φˆ12
φˆ21
φˆ22
γ̂
ĉ
ω̂
α̂
β̂
Heng Seng
Nikkei
Unadjusted
Adjusted
Unadjusted
Adjusted
Unadjusted
Adjusted
1.7833**
1.1204**
-0.2862**
-1.3476**
2.4515**
0.8634**
1.17e-6
0.0753**
0.9166**
-1.7092**
2.1950**
1.5478**
-0.2446**
2.9175**
0.1849**
3.87e-7**
0.0378**
0.9579**
-0.2238**
2.0523**
0.2260**
-1.8170**
4.3080**
-0.0162
7.08e-6**
0.1488**
0.8394**
-0.5797**
-4.4924**
0.6832**
-4.9560**
15.6068**
-0.1252**
5.44e-6**
0.1022**
0.8750**
-0.2796**
-0.1930**
0.0013**
0.0126
1.6278**
-2.0104**
3.22e-6**
0.1475**
0.8519**
-0.6837**
0.8175**
0.3890**
0.1554**
1.0431**
0.7372**
2.05e-6**
0.1017**
0.8909**
As shown in Table 3, φ12 exceeds 1 using both the adjusted and unadjusted S&P data, which suggests that the first regime follows a non-stationary
process. The same is also true for Heng Seng. However, as no results on nonstationary STAR-type models are available, it is difficult to interpret these
estimates.
In all three cases, the values of α̂ decreased and those of β̂ increased when
the data were adjusted, which agrees with other findings in the literature
22
(see, for example, Verhoeven and McAleer (1999)). However, the effects of
extreme observations and outliers on the estimates of STAR are not entirely
clear, though it appears that, if such data have a positive (negative) effect on
ĉ, they will have a negative (positive) effect on γ̂. This is an unusual result
as there is no obvious reason why the threshold value should be related to
the transition rate. However, for S&P and Heng Seng, the threshold values
are closer to 0 with the adjusted data, suggesting that the adjusted data
exhibit asymmetric behaviour. Moreover, it appears that if φˆ12 increases
(decreases) after the data are adjusted, then φˆ22 will also increase (decrease),
which suggests that extreme observations and outliers have the same effects
on the coefficients of yt−1 in both regimes.
A similar analysis is conducted to examine the effects of extreme observations and outliers for ESTAR-GARCH. The estimates are given in Table
4.
Table 4: ESTAR-GARCH Estimates for Adjusted and Unadjusted Data
S&P
φˆ11
φˆ12
φˆ21
φˆ22
γ̂
ĉ
ω̂
α̂
β̂
Heng Seng
Nikkei
Unadjusted
Adjusted
Unadjusted
Adjusted
Unadjusted
Adjusted
-0.7497**
0.0790
0.0008**
0.0360
0.9474**
-3.0813**
1.22e-6**
0.0767**
0.9149**
-0.5490**
-2.3958**
1.8336**
0.9974**
0.7736**
-0.5821**
3.87e-7**
0.0378**
0.9579**
-0.9955**
-0.9307**
0.0013**
0.1269**
0.7997**
-4.3532**
7.27e-7**
0.1506**
0.8370**
-0.4233**
0.0004
0.0013**
0.1126**
1.2412**
-2.5382**
6.07e-6**
0.1076**
0.8669**
-0.4259**
0.1168**
0.0011**
0.0137
1.6688**
-2.1846**
3.22e-6**
0.1475**
0.8519**
-0.2144**
0.6474**
0.3901**
0.1911**
0.8317**
0.7273**
2.05e-6**
0.1017**
0.8909**
The estimates of ESTAR-GARCH using the unadjusted data seem more
plausible than those for LSTAR-GARCH in Table 3. All the regimes follow a stationary AR(1) process, but the low threshold values suggest that
the second regime would dominate the first for all three indices. Furthermore, the estimates for the GARCH component are very similar to those for
23
LSTAR-GARCH and, for Nikkei, the GARCH estimates are identical! This
suggests that the choice of transition function for the conditional mean does
not affect the GARCH estimates, reflecting the block-diagonal nature of the
information matrix for STAR-GARCH models.
The effects of extreme observations and outliers on the transition rate,
γ̂, and the threshold value, ĉ, for ESTAR-GARCH seem to be different from
LSTAR-GARCH. In particular the inverse relationship between γ̂ and ĉ is
no longer valid. However, the estimated threshold values increased when the
adjusted data were used. For all three data sets, the estimated threshold
values using the adjusted data were closer to 0 than for the unadjusted data.
Furthermore, the sign of the estimated threshold value changed from negative
to positive for Nikkei.
However, the effects of extreme observations and outliers on the estimates
of the transition rates are unclear. The effects of such data on φˆ12 and φˆ22 for
ESTAR-GARCH are different from LSTAR-GARCH. In this case, if extreme
observations and outliers have positive (negative) effects on φˆ12 , they have
negative (positive) effects on φˆ22 for both S&P and Heng Seng. However,
extreme observations and outliers have negative effects on both φˆ12 and φˆ22
for Nikkei.
There does not seem to be a clear pattern between the estimates using the
adjusted and unadjusted data. However, due to the increase in the threshold
value, the second regime no longer dominates the first, so that the adjusted
data exhibit regime switching behaviour. Empirical evidence suggests that
the estimate of the threshold value is sensitive to the sign and magnitude of
outliers. If the magnitude of positive outliers is greater (smaller) than their
negative counterparts, the threshold estimate will be greater (smaller) than
0. This result explains the low threshold estimates, because the magnitude
of the negative outliers is often larger than the positive outliers for all three
indices, and also the increase in the threshold estimates when the magnitude
of the outliers is reduced.
These results also show that the effects of outliers and extreme observations on the estimates of STAR-GARCH are sensitive to the choice of
transition function. Moreover, the convergence of the estimates also seems
24
to be sensitive to the choice of transition function for STAR-STGARCH.
The choice of transition function, and their convergence of the algorithm,
are summarised in Table 5.
Table 5: Convergence with Different Transition Functions
Transition function
Unadjusted data
mean
variance
S&P
Heng Seng Nikkei
Logistic
Logistic
Yes
Yes
No
Logistic
Exponential
No
Yes
No
Exponential
Logistic
Yes
Yes
Yes
Exponential
Exponential
No
No
No
As Table 5 shows, the exponential/logistic combination converges for all
three indices. The estimates for the three data sets are given in Table 6.
The estimates of the extreme threshold values, which indicate that one
regime dominates the other, suggest that two regimes are unnecessary for
either the conditional mean or the conditional variance for S&P and Nikkei.
For the same reason, two regimes for the conditional variance are also unnecessary for Heng Seng. However, the estimated threshold value of the
conditional mean, ĉ, for Heng Seng is close to 0, which indicates the data exhibit asymmetric behaviour and two regimes are present for the conditional
mean.
Although all the estimates are highly significant, this is based on the assumption of asymptotic normality, which cannot be checked as no regularity
conditions are available. Moreover, the estimates from a single regime are
similar to those using an AR(1)-GARCH(1,1) model for S&P and Nikkei,
which reinforces the conclusion that two regimes are not present.
25
Table 6: ESTAR-LSTARGARCH Estimates for S&P, Heng Seng and Nikkei
φˆ11
φˆ12
φˆ21
φˆ22
γ̂
ĉ
ωˆ1
αˆ1
βˆ1
ωˆ2
αˆ2
βˆ2
ξˆ
dˆ
S&P
Heng Seng
Nikkei
-0.4167**
0.0862**
0.0007**
0.0359**
1.5140**
-2.3757**
1.184e-06**
0.7662**
0.9149**
4.362e-03**
0.1186**
0.00008464
3.1153**
2.9595**
-0.1124**
-0.6694**
0.8880**
-1.8741**
2.3182**
-0.2280**
6.88e-6**
0.1488**
0.8394**
1.24e-3**
0.0170**
0.0096
2.7183**
2.5338**
1.5339**
0.0991**
-0.7509**
1.2881**
0.3103**
-1.8928**
3.15e-06**
0.1465**
0.8529**
3.896e-03**
0.0331**
0.035**
3.1029**
2.9643**
Estimates for ESTAR-LSTGARCH for two sets of adjusted data are given
in Table 7, because the Newton method did not converge using the adjusted
Heng Seng data. However, the estimates did converge using the BFGS algorithm, which supports the findings in Section 3 that the estimates are
sensitive to the choice of algorithm.
It appears that extreme observations and outliers have little impact on
ˆ which suggests there is no regime switching behaviour in the GARCH
d,
component for either adjusted or unadjusted data. For each data set, only
the first regime is required. Interestingly, αˆ1 decreased and βˆ1 increased after
the data were adjusted. The same outcome holds for αˆ2 and βˆ2 for S&P, but
not for Nikkei.
Furthermore, the effects of extreme observations and outliers on γ̂ and
ĉ are also unclear. Both φˆ12 and φˆ22 decreased for S&P and Nikkei when
the adjusted data were used, which is surprising as the effects of extreme
observations and outliers for ESTAR-STGARCH are expected to be similar
26
to those for ESTAR-GARCH. This result may arise because the information
matrix of STAR-STGARCH is no longer block diagonal with regard to the
parameters of the conditional mean and the conditional variance.
For S&P, two regimes were required before the data were adjusted, but
only the second regime is significant for the adjusted data. The estimated
threshold value increased for Nikkei when the adjusted data were used, but
the second regime was still insignificant.
Table 7: ESTAR-LSTGARCH Estimates for the Adjusted S&P and Nikkei
S&P
φˆ11
φˆ12
φˆ21
φˆ22
γ̂
ĉ
ωˆ1
αˆ1
βˆ1
ωˆ2
αˆ2
βˆ2
ξˆ
dˆ
6
Nikkei
-0.0590** -0.7316**
-0.8052** -0.2209**
1.5970** 0.0013**
0.0017
0.0108
1.7790** 1.5160**
-0.1433** -2.1709**
3.75e-7** 2.00e-6**
0.0378** 0.1021**
0.9579** 0.8906**
1.10e-3** 2.38e-3**
0.0363** 0.0687**
0.0154** 0.0080**
2.9537** 2.9638**
2.7692** 2.8433**
Conclusion
This paper provided a survey of recent developments for analysing the GARCH,
STAR, STAR-GARCH and STAR-STGARCH models. The difficulties in
evaluating these models because of the absence of structural and statistical
properties, particularly, the regularity conditions for consistency and asymptotic normality, were emphasized.
27
Empirical evidence using the S&P, Heng Seng and Nikkei indexes showed
that the QMLE for STAR-GARCH models are sensitive to the choice of
optimisation algorithm. This does not agree with previous results in the
literature. It was also shown that the estimates for STAR-GARCH and
STAR-STGARCH are also highly sensitive to extreme observations and outliers. Furthermore, the effects of extreme observations and outliers on the
estimates of STAR-GARCH depend on the choice of transition function.
The effects of extreme observations and outliers on the estimates for
STAR-STGARCH are unclear, but the effects are not the same as for STARGARCH, which may arise because the information matrix is no longer block
diagonal. Furthermore, the convergence of the estimates is sensitive to the
choice of algorithm. This sensitivity could arise through model misspecification, as well as through the properties of the log-likelihood functions.
28
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